Finding The Velocity Of A Gas

Root Mean Square Velocity: We know that gases are made up of a large number of molecules.
Now, the root mean square velocity of the gas is defined as the square root of the mean of the
squares of the velocity of a large number of molecules present in the same gas. It is
represented by the letter 'u'.

u= √[u12 + u22 +
u32 -----------un2/ n]

where u1, u2, u3 are the velocities of the gas molecules.

n is the number of molecules in the gas.

Root mean square velocity or the RMS velocity is directly proportional to square root of
absolute temperature and inversely proportional to the square root of molecular weight and density.

A relationship exists between the RMS velocity and the average velocity. It is given as

Average velocity (v) = 0.9213 x RMS velocity.

Most Probable velocity: The velocity of the gas molecules keeps changing due
to frequent collisions. However, certain portion of the gas molecules have the same
velocity. Velocity possessed by a maximum number of gas molecules is termed as most
probable velocity. It is denoted by α.

A = √(2RT / M)

= √(2RT/ N x m)
( as M = N x m)

RMS velocity can also be related to the most probable velocity by a relationship.

All the three velocities RMS, Average and the Most probable velocities can be
related by the equation

α : v : u = 1: 1.128 : 1.234

Solved problems

Velocity possessed by a maximum number of molecules in the gas is termed as

Root mean square velocity

Average velocity

Most probable velocity

Any of the above.

Answer: c

Relationship between RMS velocity and the Average velocity is given by

0.9213 x RMS velocity

0.816 x RMS velocity

They cannot be related

Both are identical

Answer: a

Relationship between RMS velocity and Most probable velocity is given by

0.9213 x RMS velocity

0.816 x RMS velocity

They cannot be related

Both are identical

Answer: b

Calculate the RMS velocity, average velocity and most probable velocity of oxygen at 27οC?

Root mean square velocity is given by the formula √3RT/M

So, R = 8.314 x 107 ergs K-1 mole-1

T= 273 + 27 = 300 K

M =32

Thus, u= √(3 x 0.8314 x 107 x 300/32)

= 1.58 x 104 cm sec-1

= 48200 cm sec-1

Average velocity (v) = 0.9213 x RMS velocity.

= 0.9213 x 48200

=44406.66 cm/sec

Most Probable velocity = 0.816 x RMS velocity

= 0.816 x 48200

= 39331.2 cm/sec

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